{"id":"0680b8bd-297f-4bfb-badb-069cdd1ab9b4","arxiv_id":"2507.18495","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Combinatorial Ricci and Calabi flows on discrete conformal structures over surfaces with boundary are proven to converge exponentially to prescribed boundary lengths, globally for two structure classes and locally for the others.","lead":"This mathematics paper introduces two \"curvature flows,\" step-by-step adjustment rules, for a family of discretized hyperbolic surfaces with boundary, and proves that under these rules the boundary lengths converge to any prescribed values. It generalizes and unifies several earlier flow constructions and supplies algorithms for building discrete hyperbolic metrics with specified boundary lengths.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Abstract promises global convergence for all six structure classes, but Theorem 1.6 and Remark 5.3 only establish it for structures (3) and (6); for (4), mixed I and III the convergence is local, and for mixed II it is conditional.","rationale":"The reader's verdict is CONDITIONAL, and I agree that conditionality is appropriate. The reader's weakest_assumption identifies the imported symmetric negative definiteness of Δ as load-bearing; that is a real dependency but it is a companion-paper result rather than an internal inconsistency of this manuscript. My primary stress-test concern is different and more direct: the abstract claims global convergence without qualification, while the paper's own Theorem 1.6 and Remark 5.3 restrict four of the six classes to local convergence and one class to a conditional statement. This is not a subtle mathematical gap but a mismatch between the advertised result and the proven theorem. The mismatch is explicitly acknowledged in Remark 5.3, which states that general-initial-data global convergence is not yet shown for structures (4), mixed I, and mixed III. Because the paper's central advertised contribution is thus stronger than its own theorem statements, the abstract must be revised. The verdict should remain CONDITIONAL rather than REJECT because the global results for structures (3) and (6) appear internally coherent given the imported rigidity and Jacobian results, and the remaining proofs are plausible; however, the paper needs substantive revision of its claims and ideally independent verification of the imported negative definiteness before it can be accepted as stated.","tokens_in":18688,"tokens_out":11446,"duration_ms":126199,"concrete_test":"Audit the abstract against Theorem 1.6 and Remark 5.3: for each structure class, verify whether Sections 3–5 prove global convergence, local convergence for small initial curvature energy, or only a conditional statement, and require the abstract to match that scope. If the abstract is amended accordingly, the overstatement is resolved; the imported negative definiteness of Δ can be checked independently by sampling eigenvalues of the 3×3 Jacobian in (19)/(21)–(23) on U(η) for structures (3) and (6).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract's unqualified claim that the paper establishes 'longtime existence and global convergence of solutions to these combinatorial curvature flows' is not supported by the theorems proved. Theorem 1.6(i) gives global exponential convergence only for structure (3) with α:B→{0,1} and η satisfying η_ij>α_iα_j, and Theorem 1.6(ii) gives it only for structure (6). For structure (4) and mixed structures I and III, Theorem 1.6(iii) asserts existence and exponential convergence only when ∥K(u(0))−Kbar∥<δ, i.e., for initial data close to a solution of K(u)=Kbar. Remark 5.3 explicitly concedes that for these classes the authors have not shown the solution cannot reach some boundary of the admissible space, so general-initial-data global convergence is unresolved. For mixed structure II, Theorem 1.6(iv) is conditional even locally: a convergent solution must have limit curvature Kbar, and local convergence is asserted only if a discrete conformal factor u with K(u)=Kbar is already known to exist. Thus the paper's stated central claim is materially stronger than what is proved. The global results that are proved additionally depend on symmetric negative definiteness of Δ imported from [17, Theorems 3.2 and 4.2] and on the rigidity/existence Theorem 1.3 from [17]; these ingredients are not re-derived here, so any gap in [17] would propagate into the global cases.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript is the third in a series on discrete conformal structures on ideally triangulated surfaces with boundary. It introduces combinatorial Ricci flow (10) and combinatorial Calabi flow (11) for the six structure classes classified in [16], and studies their longtime behavior using energy functions E and C, the negative definiteness of the curvature Jacobian imported from [17], and boundary-avoidance estimates. The main result, Theorem 1.6, proves global exponential convergence for structures (3) and (6) from arbitrary initial data in the admissible space; for structures (4) and mixed I and III it proves exponential convergence only under a smallness assumption on the initial curvature deviation; for mixed II it proves a conditional statement. Remark 5.3 acknowledges that global results for the latter classes are not established.","tokens_in":18916,"tokens_out":15033,"duration_ms":147246,"significance":"The paper's strengths are the detailed boundary-avoidance estimates for the global cases (Lemmas 3.4-3.6, 3.9-3.12, 4.4-4.12), the clean Lyapunov-function structure, and the unifying character of the flows, which specialize to several previously studied flows (Guo-Luo, Guo, Li-Xu-Zhou, Luo-Xu, Xu). If correct, it yields practical algorithms for finding discrete hyperbolic metrics with prescribed boundary lengths. The significance is real but qualified: half of the six structure classes are only handled locally or conditionally, and the global cases rest on the symmetric negative definiteness of Delta and on Theorem 1.3, both taken from the companion preprint [17].","major_comments":[{"comment":"The abstract claims 'longtime existence and global convergence of solutions to these combinatorial curvature flows' without qualification, and §1.1 repeats this. This is materially stronger than what is proved: Theorem 1.6(iii) gives existence and exponential convergence only when ||K(u(0))-K|| < delta for structures (4) and mixed I/III, Theorem 1.6(iv) is conditional for mixed II, and Remark 5.3 states explicitly that the authors have not shown that solutions for these classes cannot reach a boundary of the admissible space. Please revise the abstract and introduction to state the global results for (3) and (6) and the local/conditional results for the remaining classes.","section":"Abstract, §1.1, and Theorem 1.6"},{"comment":"The proofs of global convergence depend essentially on the strict convexity of E and the symmetric negative definiteness of the Jacobian Delta, which are quoted from the companion preprint [17] (Theorems 3.2 and 4.2), and on the rigidity/existence Theorem 1.3 also from [17]. These facts are not re-derived here. Because [17] is cited as an arXiv preprint (arXiv:2407.19501v3), the present results are conditional on an unpublished source; the authors should either include the precise statements and key Jacobian computations needed, or clearly mark this dependence and ensure that [17] is accessible to the reader. This is a verifiability issue that should be addressed before publication.","section":"§2 (after Eqs. (12)-(13)) and Theorems 3.2, 4.2"}],"minor_comments":[{"comment":"The prescribed curvature and the actual curvature are both denoted by K; for instance, formulas (10) and (11) read 'du_i/dt = K_i - K_i'. Use a distinct symbol such as \\bar K for the target curvature.","section":"Throughout, Eqs. (10)-(11)"},{"comment":"There are repeated typos: 'weighs' should be 'weights' in Theorem 3.1, Definition 3.3, and Section 3; also '1.3 (i)' in the proof of Theorem 3.8 should be 'Theorem 1.3(i)'.","section":"§3, Theorem 3.1, Definition 3.3"},{"comment":"The transition from convergence along the sequence xi_n to convergence of the full trajectory is not spelled out; the authors should justify it, for example by noting that for large n the point u(xi_n) enters the basin of attraction given by the Lyapunov Stability Theorem, or by a direct strict-convexity argument for E.","section":"Theorem 3.8"},{"comment":"State explicitly that initial values u(0) are required to lie in the admissible space U(eta).","section":"Definition 1.4 and Theorem 1.6"},{"comment":"The sign 'theta^{jk}_i -> 0^-' appears to be a typo; boundary arc lengths are positive, so the limit should be 0^+.","section":"Lemmas 3.5 and 4.6"}],"recommendation":"major_revision","confidential_remarks":"The paper is part of a series, and the main global convergence results rely on the companion preprint [17], which is not yet published. If the journal does not normally accept results conditional on unpublished preprints, this should be discussed with the authors. The abstract overstatement in the current version should be corrected before acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: the paper proves what the abstract claims only for two of the six structure classes. For structures (3) and (6), the combinatorial Ricci and Calabi flows are shown to exist globally and converge exponentially from any initial data, and those proofs look credible. For structures (4), mixed I, II, and III, the theorems are local (close to a known solution) or conditional, and Remark 5.3 explicitly concedes that global behavior near the boundary is unresolved. The abstract says unqualified “longtime existence and global convergence,” which is materially stronger than what is proved.\n\nThe genuinely new thing is the unified setup: the authors define combinatorial Ricci and Calabi flows covering all six discrete conformal structure types from their classification, including the mixed types, and they show how the new flows reduce to the known flows of Guo–Luo, Guo, Li–Xu–Zhou, Luo–Xu, and Xu in special cases. That is a real contribution and will be useful to anyone working on circle packings or discrete conformal geometry with boundary.\n\nThe technical work for the global cases is careful. The boundary estimates in Lemmas 3.4–3.6, 3.10–3.12, and 4.4–4.12 are detailed and do the heavy lifting: they show solutions cannot hit the boundary of the admissible space, so the flow stays in a compact set. The negative-definiteness of the curvature Laplacian then gives exponential convergence. For structures (3) and (6), the proof is standard but cleanly executed, and I would trust those theorems.\n\nThe soft spots are exactly where the scope shrinks. Several main theorems (4.8, 4.14, 5.2(2)) are stated with proofs omitted as “analogous.” If they really are analogous, that is acceptable, but a referee will have to verify it. The negative-definiteness of the Jacobian is imported from the companion paper [17] without re-derivation; that is fine in a series, but it means a gap in [17] would propagate here. The main issue is that the paper presents itself as solving the deformation problem for all six structures when it really solves two globally and gives local results for the rest. The authors are honest about this in Remark 5.3, but the abstract is not.\n\nWho is this for? Specialists in discrete conformal geometry and combinatorial curvature flows. A reader in that area gets value from the unified framework and the two global convergence theorems. For a more general differential geometer, the scope overstatement could mislead, but the underlying mathematics is serious.\n\nRecommendation: send it to peer review. The referee should ask for the abstract to be aligned with the theorems and for at least the most important omitted proofs to be included or clearly deferred. The global theorems for (3) and (6) are strong enough to justify referee time.","headline":"Solid global convergence proofs for two structure classes, but the abstract overclaims; the rest is local or conditional and should be labeled as such.","tokens_in":19507,"tokens_out":1489,"would_cite":true,"duration_ms":16570,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["52C25","52C26"],"pacs":[],"model":"deepseek-v4-flash","headline":"Combinatorial Ricci and Calabi flows converge exponentially on surfaces with boundary, realizing prescribed boundary lengths.","keywords":["combinatorial Ricci flow","combinatorial Calabi flow","discrete conformal structures","surfaces with boundary","hyperbolic surfaces with totally geodesic boundary","prescribed boundary lengths","exponential convergence","curvature flows"],"falsifier":"Take the simplest ideally triangulated surface with boundary—one ideal face giving a single right-angled hyperbolic hexagon with three boundary components—and compute the $3\\times 3$ Jacobian $\\partial \\theta/\\partial u$ numerically along a path in $U(\\eta)$ approaching the boundary $u_i+u_j = C(\\eta_{ij})$. If any eigenvalue reaches zero or the matrix loses symmetry before the boundary, the negative-definiteness premise fails and the global convergence claim for that structure collapses. Alternatively, for structure (4), run the Ricci flow from initial data with large $\\|K(u(0))-\\overline{K}\\|$ and check whether $u_i$ reaches $-\\pi/2$ in finite time.","tokens_in":18389,"feed_emoji":"📐","tokens_out":8542,"duration_ms":76493,"temperature":0.7,"pith_summary":"This paper extends combinatorial curvature flow theory from closed surfaces to surfaces with boundary. It introduces a combinatorial Ricci flow and a combinatorial Calabi flow for the six families of discrete conformal structures classified in the authors' earlier work. The main theorem states that for structures of types (3) and (6), these flows exist for all time from any starting point and converge exponentially fast to the unique discrete conformal factor realizing any prescribed positive boundary lengths; for structures (4) and the mixed types I and III, the same conclusion holds when the initial curvature is within a small distance of the target. Because a discrete hyperbolic metric is determined by its boundary lengths, this turns the flow convergence into an algorithm for constructing hyperbolic surfaces with totally geodesic boundaries of prescribed lengths.","feed_headline":"Ricci and Calabi flows converge exponentially on bordered surfaces","feed_subtitle":"Two discrete curvature flows deform surface boundary lengths to prescribed values, proving longtime existence and convergence.","key_machinery":"The load-bearing object is the Jacobian $\\Delta = (\\partial K_i/\\partial u_j)$ of the generalized combinatorial curvature map with respect to the transformed variables $u_i$, restricted to the convex admissible space $U(\\eta) = \\bigcap_{\\{ijk\\}\\in F} U_{ijk}(\\eta)$. From the companion paper [17], this matrix is symmetric and negative definite everywhere on $U(\\eta)$. That single fact does three jobs: it makes $E(u)$ and $C(u)$ strictly convex Lyapunov functions, it guarantees at most one solution of $K(u)=\\overline{K}$, and it supplies a uniform spectral gap yielding exponential decay of the curvature error. Boundary non-reachability is handled by separate lemmas showing that the flows cannot hit infinity, the $u_i = 0$ boundary, or the degeneration boundary where an ideal edge length $l_{ij}$ vanishes.","core_discovery":"For an ideally triangulated compact surface with boundary carrying one of the discrete conformal structures (3), (4), (6), mixed I, II, or III from Theorem 1.1, define the generalized combinatorial curvature $K_i$ as the total boundary-arc length at boundary component $i$. The combinatorial Ricci flow $du_i/dt = K_i - \\overline{K}_i$ and the combinatorial Calabi flow $du_i/dt = -\\Delta(K-\\overline{K})_i$ are negative gradient flows of the strictly convex energy $E(u)$ and the curvature energy $C(u)$. Theorem 1.6 asserts: for structure (3) with $\\alpha \\in \\{0,1\\}$ and $\\eta_{ij} > \\alpha_i\\alpha_j$, and for structure (6) with $\\eta > 0$, both flows exist for all time and converge exponentially fast for every initial value $u(0)$; for structure (4) and mixed structures I and III, there is a $\\delta > 0$ such that $\\|K(u(0))-\\overline{K}\\| < \\delta$ implies longtime existence and exponential convergence; for mixed structure II, any convergent solution must satisfy $K(u) = \\overline{K}$, and if such a solution exists then nearby initial data converge exponentially. The abstract states global convergence; the precise theorem limits the general-initial-data claim to structures (3) and (6).","pith_inferences":["The same Lyapunov-function mechanism would likely transfer to fractional or higher-order curvature flows on surfaces with boundary, since the argument uses only negative definiteness and a spectral gap; the paper does not pursue this extension.","For structure (4) and mixed structure III, the missing ingredient for global-initial-data convergence is a uniform lower bound away from the boundary of the admissible space; a numerical experiment starting far from the target would show whether the small-initial-curvature condition is merely technical or reflects actual escape.","Because the flows are ODEs in finitely many variables, the proof immediately suggests an explicit time-stepping algorithm, though no implementation or numerical experiments are reported here.","The exponential convergence gives a discrete analogue of the rigidity statement that hyperbolic surfaces with totally geodesic boundary are determined by their boundary lengths, which could serve as a benchmark for numerical discretizations of hyperbolic surfaces."],"forward_implications":["For structures (3) and (6), prescribed positive boundary lengths are always realized by a unique discrete conformal factor, and both flows find it from any starting value.","The exponential convergence rate is uniform along the flow, so the corresponding algorithms terminate in finite time to any prescribed tolerance.","These flows unify and generalize previously known combinatorial Yamabe, Ricci, and Calabi flows for surfaces with boundary, recovering them as special cases under specific choices of the parameters.","For structures (4) and mixed I and III, the target curvature configuration is a local attractor: initial curvature errors below $\\delta$ lie in its basin of exponential convergence.","For mixed structure II, a necessary condition is established: a convergent flow necessarily converges to a solution of $K(u)=\\overline{K}$; existence of such a solution makes convergence local."],"supporting_citations":[{"why":"Supplies the rigidity and existence theorem for the discrete conformal structures and the symmetric negative-definite Jacobian results on which the Lyapunov arguments rest.","marker":"[17]"},{"why":"Classifies the six discrete conformal structure types and gives the explicit formulas (3)-(9) that the flows deform.","marker":"[16]"},{"why":"Originates the combinatorial Ricci flow framework for circle packings that the present flows generalize to surfaces with boundary.","marker":"[1]"},{"why":"Provides the generalized circle-packing curvature flow on bordered surfaces and the limiting-angle lemma used to keep flows away from a degeneracy boundary.","marker":"[6]"},{"why":"Supplies the Lyapunov Stability Theorem invoked to convert a local attractor plus compactness into exponential convergence.","marker":"[10]"},{"why":"Earlier flow-convergence result for a special class of generalized circle packings on surfaces with boundary that the present theorem generalizes.","marker":"[15]"}],"fun_headline_variants":["Discrete Ricci and Calabi flows converge on bordered surfaces","Boundary lengths prescribed by discrete curvature flows","Two convergent flows target surface boundary lengths","Curvature flows deform boundary lengths to prescribed values","Bordered surfaces: discrete flows converge to target lengths"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole proof leans on one imported fact: at every allowed choice of the parameters, the matrix recording how each boundary curvature responds to each parameter change is symmetric and negative definite; if that property failed somewhere in the admissible region, the Lyapunov functions would not be convex and the flows could leave the allowable region or fail to converge.","fun_headline_variants_meta":{"raw":{"variants":["Discrete Ricci and Calabi flows converge on bordered surfaces","Boundary lengths prescribed by discrete curvature flows","Two convergent flows target surface boundary lengths","Curvature flows deform boundary lengths to prescribed values","Bordered surfaces: discrete flows converge to target lengths"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001035,"raw_usage":{"total_tokens":4355,"prompt_tokens":939,"completion_tokens":3416,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":555,"completion_tokens_details":{"reasoning_tokens":3344}},"tokens_in":555,"tokens_out":3416,"duration_ms":29528,"temperature":1.0,"reasoning_tokens":3344,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T18:12:29.546862+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the simplest ideally triangulated surface with boundary—one ideal face giving a single right-angled hyperbolic hexagon with three boundary components—and compute the $3\\times 3$ Jacobian $\\partial \\theta/\\partial u$ numerically along a path in $U(\\eta)$ approaching the boundary $u_i+u_j = C(\\eta_{ij})$. If any eigenvalue reaches zero or the matrix loses symmetry before the boundary, the negative-definiteness premise fails and the global convergence claim for that structure collapses. Alternatively, for structure (4), run the Ricci flow from initial data with large $\\|K(u(0))-\\overline{K}\\|$ and check whether $u_i$ reaches $-\\pi/2$ in finite time.","supporting_citations":[{"cited_title":"Discrete conformal structures on surfaces with boundary (II) -- Rigidity and Existence","cited_arxiv_id":"2407.19501","evidence_quote":"Supplies the rigidity and existence theorem for the discrete conformal structures and the symmetric negative-definite Jacobian results on which the Lyapunov arguments rest."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Classifies the six discrete conformal structure types and gives the explicit formulas (3)-(9) that the flows deform."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Originates the combinatorial Ricci flow framework for circle packings that the present flows generalize to surfaces with boundary."},{"cited_title":"Pontryagin, Ordinary differential equations , Addison-Wesley Publishing Company Inc., Reading, 1962","cited_arxiv_id":null,"evidence_quote":"Supplies the Lyapunov Stability Theorem invoked to convert a local attractor plus compactness into exponential convergence."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Earlier flow-convergence result for a special class of generalized circle packings on surfaces with boundary that the present theorem generalizes."}],"review_version":2}